dihedral group presentation proof

This is a group presentation whose rigorous definition has something to do with quotient groups of free groups, or some such. However, I am only given the following non-rigorous definition:

is the group generated by the objects and , which are known to satisfy the following relations: and (where is the identity).

I'm then asked to find the order of the cyclic subgroup generated by . In other words, what is the order of ?

I just don't know how to make this work. Obviously the answer is , but I don't know how to prove that from the available information. In fact, I suspect I may have somehow missed something. You can download a pdf of the textbook chapter here (the entire book can be had here, but it's in an unusual format called *.djvu). The exercise in question is #8 from that section (1.2).